309.80/291.49	WORST_CASE(Omega(n^1), ?)
309.80/291.50	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
309.80/291.50	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
309.80/291.50	
309.80/291.50	
309.80/291.50	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
309.80/291.50	
309.80/291.50	(0) CpxTRS
309.80/291.50	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
309.80/291.50	(2) TRS for Loop Detection
309.80/291.50	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
309.80/291.50	(4) BEST
309.80/291.50	    (5) proven lower bound
309.80/291.50	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
309.80/291.50	        (7) BOUNDS(n^1, INF)
309.80/291.50	    (8) TRS for Loop Detection
309.80/291.50	
309.80/291.50	
309.80/291.50	----------------------------------------
309.80/291.50	
309.80/291.50	(0)
309.80/291.50	Obligation:
309.80/291.50	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
309.80/291.50	
309.80/291.50	
309.80/291.50	The TRS R consists of the following rules:
309.80/291.50	
309.80/291.50	   cond1(true, x, y, z) -> cond2(gr(y, z), x, y, z)
309.80/291.50	   cond2(true, x, y, z) -> cond2(gr(y, z), p(x), p(y), z)
309.80/291.50	   cond2(false, x, y, z) -> cond1(and(eq(x, y), gr(x, z)), x, y, z)
309.80/291.50	   gr(0, x) -> false
309.80/291.50	   gr(s(x), 0) -> true
309.80/291.50	   gr(s(x), s(y)) -> gr(x, y)
309.80/291.50	   p(0) -> 0
309.80/291.50	   p(s(x)) -> x
309.80/291.50	   eq(0, 0) -> true
309.80/291.50	   eq(s(x), 0) -> false
309.80/291.50	   eq(0, s(x)) -> false
309.80/291.50	   eq(s(x), s(y)) -> eq(x, y)
309.80/291.50	   and(true, true) -> true
309.80/291.50	   and(false, x) -> false
309.80/291.50	   and(x, false) -> false
309.80/291.50	
309.80/291.50	S is empty.
309.80/291.50	Rewrite Strategy: FULL
309.80/291.50	----------------------------------------
309.80/291.50	
309.80/291.50	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
309.80/291.50	Transformed a relative TRS into a decreasing-loop problem.
309.80/291.50	----------------------------------------
309.80/291.50	
309.80/291.50	(2)
309.80/291.50	Obligation:
309.80/291.50	Analyzing the following TRS for decreasing loops:
309.80/291.50	
309.80/291.50	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
309.80/291.50	
309.80/291.50	
309.80/291.50	The TRS R consists of the following rules:
309.80/291.50	
309.80/291.50	   cond1(true, x, y, z) -> cond2(gr(y, z), x, y, z)
309.80/291.50	   cond2(true, x, y, z) -> cond2(gr(y, z), p(x), p(y), z)
309.80/291.50	   cond2(false, x, y, z) -> cond1(and(eq(x, y), gr(x, z)), x, y, z)
309.80/291.50	   gr(0, x) -> false
309.80/291.50	   gr(s(x), 0) -> true
309.80/291.50	   gr(s(x), s(y)) -> gr(x, y)
309.80/291.50	   p(0) -> 0
309.80/291.50	   p(s(x)) -> x
309.80/291.50	   eq(0, 0) -> true
309.80/291.50	   eq(s(x), 0) -> false
309.80/291.50	   eq(0, s(x)) -> false
309.80/291.50	   eq(s(x), s(y)) -> eq(x, y)
309.80/291.50	   and(true, true) -> true
309.80/291.50	   and(false, x) -> false
309.80/291.50	   and(x, false) -> false
309.80/291.50	
309.80/291.50	S is empty.
309.80/291.50	Rewrite Strategy: FULL
309.80/291.50	----------------------------------------
309.80/291.50	
309.80/291.50	(3) DecreasingLoopProof (LOWER BOUND(ID))
309.80/291.50	The following loop(s) give(s) rise to the lower bound Omega(n^1):
309.80/291.50	
309.80/291.50	The rewrite sequence
309.80/291.50	
309.80/291.50	gr(s(x), s(y)) ->^+ gr(x, y)
309.80/291.50	
309.80/291.50	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
309.80/291.50	
309.80/291.50	The pumping substitution is [x / s(x), y / s(y)].
309.80/291.50	
309.80/291.50	The result substitution is [ ].
309.80/291.50	
309.80/291.50	
309.80/291.50	
309.80/291.50	
309.80/291.50	----------------------------------------
309.80/291.50	
309.80/291.50	(4)
309.80/291.50	Complex Obligation (BEST)
309.80/291.50	
309.80/291.50	----------------------------------------
309.80/291.50	
309.80/291.50	(5)
309.80/291.50	Obligation:
309.80/291.50	Proved the lower bound n^1 for the following obligation:
309.80/291.50	
309.80/291.50	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
309.80/291.50	
309.80/291.50	
309.80/291.50	The TRS R consists of the following rules:
309.80/291.50	
309.80/291.50	   cond1(true, x, y, z) -> cond2(gr(y, z), x, y, z)
309.80/291.50	   cond2(true, x, y, z) -> cond2(gr(y, z), p(x), p(y), z)
309.80/291.50	   cond2(false, x, y, z) -> cond1(and(eq(x, y), gr(x, z)), x, y, z)
309.80/291.50	   gr(0, x) -> false
309.80/291.50	   gr(s(x), 0) -> true
309.80/291.50	   gr(s(x), s(y)) -> gr(x, y)
309.80/291.50	   p(0) -> 0
309.80/291.50	   p(s(x)) -> x
309.80/291.50	   eq(0, 0) -> true
309.80/291.50	   eq(s(x), 0) -> false
309.80/291.50	   eq(0, s(x)) -> false
309.80/291.50	   eq(s(x), s(y)) -> eq(x, y)
309.80/291.50	   and(true, true) -> true
309.80/291.50	   and(false, x) -> false
309.80/291.50	   and(x, false) -> false
309.80/291.50	
309.80/291.50	S is empty.
309.80/291.50	Rewrite Strategy: FULL
309.80/291.50	----------------------------------------
309.80/291.50	
309.80/291.50	(6) LowerBoundPropagationProof (FINISHED)
309.80/291.50	Propagated lower bound.
309.80/291.50	----------------------------------------
309.80/291.50	
309.80/291.50	(7)
309.80/291.50	BOUNDS(n^1, INF)
309.80/291.50	
309.80/291.50	----------------------------------------
309.80/291.50	
309.80/291.50	(8)
309.80/291.50	Obligation:
309.80/291.50	Analyzing the following TRS for decreasing loops:
309.80/291.50	
309.80/291.50	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
309.80/291.50	
309.80/291.50	
309.80/291.50	The TRS R consists of the following rules:
309.80/291.50	
309.80/291.50	   cond1(true, x, y, z) -> cond2(gr(y, z), x, y, z)
309.80/291.50	   cond2(true, x, y, z) -> cond2(gr(y, z), p(x), p(y), z)
309.80/291.50	   cond2(false, x, y, z) -> cond1(and(eq(x, y), gr(x, z)), x, y, z)
309.80/291.50	   gr(0, x) -> false
309.80/291.50	   gr(s(x), 0) -> true
309.80/291.50	   gr(s(x), s(y)) -> gr(x, y)
309.80/291.50	   p(0) -> 0
309.80/291.50	   p(s(x)) -> x
309.80/291.50	   eq(0, 0) -> true
309.80/291.50	   eq(s(x), 0) -> false
309.80/291.50	   eq(0, s(x)) -> false
309.80/291.50	   eq(s(x), s(y)) -> eq(x, y)
309.80/291.50	   and(true, true) -> true
309.80/291.50	   and(false, x) -> false
309.80/291.50	   and(x, false) -> false
309.80/291.50	
309.80/291.50	S is empty.
309.80/291.50	Rewrite Strategy: FULL
309.80/291.53	EOF